*by Tanguy Marchand, Luc Blanchet and Guillame Faye.*

With the spectacular discoveries by the LIGO/VIRGO collaboration of gravitational waves from the coalescence of black-hole binaries, we foresee the possibility of extremely accurate measurements of the so-called post-Newtonian (PN) coefficients that describe the gravitational waveform of these systems in the inspiral phase prior to the final coalescence. The PN coefficients are especially important because they probe the non-linear structure of general relativity (GR) and provide thus very constraining tests of this theory. In turn, they permit accurate measurements of the physical parameters of the binary, essentially the mass of the compact objects and their moment of rotation or spin.

The PN coefficients mainly enter the waveform through the expression of the binary’s orbital phase, whose time evolution is driven by gravitational radiation reaction. To Newtonian order, this evolution is basically obtained from the Einstein quadrupole formula of 1918. Now, after years of theoretical developments on approximation methods in GR, the computation of the phase evolution has been achieved up to the 3.5PN order, which corresponds to the order (*v/c*)^{7 }beyond the Newtonian quadrupole formula, where* v* is the orbital velocity and *c* is the speed of light.

The LIGO/VIRGO collaboration has already published some upper bounds on possible deviations of PN coefficients based on the combined observations of the gravitational-wave events GW150914 and GW151226, as shown in Fig. 2. In particular, GW151226 (i.e., the event from December 26, 2015), because of the lower black-hole masses involved, gives acces to a larger portion of the inspiral phase, with about 50 orbital cycles, allowing us to better test the PN expansion. We expect that, thanks to future observations of neutron-star binaries, with tens of thousands of orbital cycles in the frequency band of detectors, very accurate tests of the PN coefficients will be possible.

At the 1.5PN order, the PN coefficient reflects the presence of a particular interaction, known as the tail effect, between the multipole moments describing the binary system. Gravitational-wave tails have been extensively investigated in the literature from the 1950’s. In the present case, they are due to the backscattering of quadrupole waves off the space-time curvature generated by the mass monopole of the source. They thus correspond to a quadratic non-linear effect. As we see from Fig. 2, this effect has been constrained by the LIGO observations and differs from the predicted GR value by at most 10 percent.

At the 3PN order, the PN coefficients start depending on cubic non-linear interactions. Those are, essentially, interactions between tails and, again, the mass of the source. Such interactions, referred to as “tails-of-tails”, have been known for a while. In our paper, we have computed the next non-linear effect, namely that of the interaction of the tails-of-tails with the mass again, which we naturally call “tails-of-tails-of-tails”. This is a quartic non-linear effect, with one quadrupole moment interacting with three mass monopoles. Of course, it involves at once interactions between simple tails and two further masses, as well as those between tails-of-tails and one further mass. We showed that the tails-of-tails-of-tails arise at the 4.5PN order in the waveform, i.e. at order (*v/c*)^{9 }beyond the quadrupole formula. As suggested by Fig. 2, the control of such high 4.5PN order is likely to become observationally relevant in the coming years!

We developed a machinery of complicated integration formulas, made of a cocktail of Legendre polynomials and associated functions. This, together with extensive Mathematica computations, has allowed us to control the full coefficient at the 4.5PN order in the gravitational-wave energy flux emitted by compact binaries. The GW energy flux is the main input required to control the orbital phase evolution, which is in turn indispensable for the construction of accurate gravitational-wave templates. Our 4.5PN result is in perfect agreement, in the test-particle limit (when the mass ratio *m _{1}/m_{2 }*goes to zero), with the result obtained by means of black-hole perturbation methods applied to the two-body problem. However, importantly, notice that the previous coefficient at the 4PN order is still unknown! Its computation requires more work, notably because we need the mass quadrupole moment of the compact binary at the 4PN order, which constitutes our next goal.

*Read the full article in Classical and Quantum Gravity: Gravitational-wave tail effects to quartic non-linear order Marchand, Blanchet and Faye 2016 Class. Quantum Grav. 33 244003*

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